| L(s) = 1 | + 2-s + 4-s + 2.30·5-s − 2.60·7-s + 8-s + 2.30·10-s + 2.30·11-s + 1.30·13-s − 2.60·14-s + 16-s + 6·17-s + 2·19-s + 2.30·20-s + 2.30·22-s − 3.90·23-s + 0.302·25-s + 1.30·26-s − 2.60·28-s + 3.90·29-s − 0.302·31-s + 32-s + 6·34-s − 6·35-s + 37-s + 2·38-s + 2.30·40-s − 9.90·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.02·5-s − 0.984·7-s + 0.353·8-s + 0.728·10-s + 0.694·11-s + 0.361·13-s − 0.696·14-s + 0.250·16-s + 1.45·17-s + 0.458·19-s + 0.514·20-s + 0.490·22-s − 0.814·23-s + 0.0605·25-s + 0.255·26-s − 0.492·28-s + 0.725·29-s − 0.0543·31-s + 0.176·32-s + 1.02·34-s − 1.01·35-s + 0.164·37-s + 0.324·38-s + 0.364·40-s − 1.54·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.626930209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.626930209\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 - 2.30T + 5T^{2} \) |
| 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 - 2.30T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 3.90T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 + 0.302T + 31T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 - 0.605T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 + 3.51T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 - 9.21T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.21504624504124819813074099459, −9.941106386784632519402725495147, −8.961346271187487164991698361971, −7.74028955646227738291857137000, −6.58726137798134571383683772350, −6.05158282859351272481720035666, −5.21146575159579199104920070104, −3.82255826082229337591182372438, −2.95468931311390124516057995731, −1.51849191635781980303548251648,
1.51849191635781980303548251648, 2.95468931311390124516057995731, 3.82255826082229337591182372438, 5.21146575159579199104920070104, 6.05158282859351272481720035666, 6.58726137798134571383683772350, 7.74028955646227738291857137000, 8.961346271187487164991698361971, 9.941106386784632519402725495147, 10.21504624504124819813074099459
